Dirac Notation Help

We are working on Dirac notation in my quantum class, and for the most part I see that it is a very easy way to do problems. But I am still getting stuck on how to deal with a few things on my current homework assignment. These come out of chapter 2 in the Cohen-Tannoudji book if you want to look them up.

#1. |[tex]\phi_{n}[/tex]> are eigenstates of a Hermitian operator H and they form a discrete orthonormal basis. The operator U(m,n) is defined by U(m,n)=|[tex]\phi_{m}[/tex]><[tex]\phi_{n}[/tex]|.

b. Calculate the commutator [H,U(m,n)].

I'm not really sure how to deal with this in such a general case. I get to the first step: [tex]H|\phi_{m}><\phi_{n}|-|\phi_{m}><\phi_{n}|H[/tex]

but I don't know where to go from there.

e. Let A be an operator, with matrix elements [tex]A_{mn}=<\phi_{m}|A|\phi_{n}>[/tex]

Prove the relation:[tex]A=\Sigma A_{mn}U(m,n)[/tex]

If I start with [tex]A_{mn}=<\phi_{m}|A|\phi_{n}>[/tex], is it legal to do this:

#1
b. If [tex]|\phi_{n}>[/tex] is a eigenket of H, then what is the action of H on [tex]|\phi_{n}>[/tex]? Similarly, what is [tex]<\phi_{n}|H[/tex]? Also note that the communtator between two operators is in general an operator.

e. The identity operator is [tex]\sum_{n}|\phi_{n}><\phi_{n}|[/tex]. Presumably you want to find A=?. Try to insert the identity operator in front and after A and exchange terms to see what you get.

#4

What is a projector? [tex]\frac{|\phi><\phi|}{<\phi|\phi>}[/tex] and [tex]\frac{|\psi><\psi|}{<\psi|\psi>}[/tex] would be projectors.

It seems to me that if you did things slightly different, you'd get the answer you seek.

I started with [tex] K=|\phi><\psi|[/tex] and multiplied by [tex]\frac{<\phi|\phi>}{<\phi|\phi>}[/tex] and [tex]\frac{<\psi|\psi>}{<\psi|\psi>}[/tex]. That is how I got the [tex]\lambda=<\phi|\psi>[/tex] term out front. Although, now that I look over it again, I'm not sure if I can rearrange the terms like that.